Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

We choose to reword this for our students. Instead of I can construct a viable argument, we say I can show my work so a reader understands having to ask me questions.

We use the following learning progression to help students self-assess and reach to deepen their learning.

Now, Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s by Judy Cox gives away the mathematical thinking on some pages. We decided to read the book and ask our students to listen and take notes as readers, writers, and mathematicians. Mathematicians notice and note details, look for patterns, and ask questions. To support listening and comprehension (a.k.a. empower learners to make sense and persevere), we created visuals for quasi-reader’s theater and spelled sheep, alpaca, llama, and yak. (Level 2; check.)

We also practiced a keep the pace up and get kids collaborating instead of relying on the teacher strategy we are learning from Elizabeth Statmore.

And every day I used 10-2 processing to keep the pace up and get kids collaborating instead of relying on me. For every ten minutes of notes, I gave two minutes of processing time to catch up and collaborate on making their notes accurate. (Statmore, n pag.)

Instead of 10-2 processing, we took a minute after every couple of pages to intentionally turn and talk with a partner with the express purpose of comparing and improving our notes and mathematical communication.

As teachers, we are striving to implement tasks that promote reasoning and problem solving. Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s is a counting book so 1st graders can tackle the math. 2nd and 3rd graders can use this to connect skip counting and repeated addition to multiplication and to use and connect mathematical representations. 4th and 5th graders can use this to use and connect mathematical representations while attending to precision. (Level 1; check.)

Here’s a messy version of how we anticipated student work and thinking.

These read-aloud moments open up the opportunity for rich discussion and engaging questions. Students have the opportunity for more organic and deeper understanding of mathematical concepts thanks to the book that brought them to life, and it is an engaging way to look at math through a different lens.

I can describe or illustrate ow I arrived at a solution so that the reader understands without talking to me?

Isn’t this really about making thinking visible and clear communication? Anyone who has taught learners who take an AP exam can attest to the importance of organized, clear pathways of thinking. It is not about watching the teacher show work, it is about practicing, getting feedback, and revising.

Compare the following:

What if a learner submits the following work?

Can the reader understand how the writer arrived at this solution without asking any questions?

What if the learner shared more thinking? Would it be clearer to the reader? What do you think?

How often do we tell learners that they need to show their work? What if they need to show more work? What if they don’t know how?

How might we communicate and collaborate creatively to show and tell how to level up in showing work and making thinking visible?

How might we grow in the areas of comprehension, accuracy, flexibility, and deeper understanding if we learn to communicate clearly using words, pictures, and numbers?

18, 27, 36, 45, 54, 63, 72Yes! How did you find the numbers to continue the pattern?

S1: I added 9.(Me: That’s what I did.)
S2: I multiplied by 9.(Me: Uh oh…)
S3: The ones go down by 1 and the tens go up by 1.(Me: Wow, good connection.)

Arleen and Laura probed and pushed for deeper explanations.

S1: To get to the next number, you always add 9.(Me: That’s what I did.)
S2: I see 2×9, 3×9, and 4×9, so then you’ll have 5×9, 6×9, 7×9, and 8×9.(Me: Oh, I see! She is using multiples of 9, not multiplying by 9. Did she mean multiples not multiply?)
S3: It’s always the pattern with 9’s.(Me: He showed the trick about multiplying by 9 with your hands.)

Without the probing and pushing for explanations, I would have thought some of the children did not understand. This is where in-the-moment formative assessment can accelerate the speed of learning.

There were several more examples with probing for understanding. Awesome work by this team to push and practice. Arleen and Laura checked in with every child as they worked to coach every learner to success. Awesome!

I was so curious about the children’s thinking. Look at the difference in their work and their communication.

By analyzing their work in the moment, we discovered that they were seeing the patterns, getting the answers, but struggled to explain their thinking. It got me thinking…How often in math do we communicate to children that a right answer is enough? And the faster the better??? Yikes! No, no, no! Show what you know, not just the final answer.

My turn to teach.

It is not enough to have the correct numbers in the answer. It is important to have the correct numbers, but that is not was is most important. It is critical to learn to describe your thinking to the reader.

How might we explain our thinking? How might we show our work? This is what your teachers are looking for.

The children gave GREAT answers!

We can write a sentence.
We can draw a picture.
We can show a number algorithm. (Seriously, a 4th grader gave this answer. WOW!)

But, telling me what I want to hear is very different than putting it in practice.

It makes me wonder… How can I communicate better to our learners? How can I show a path to successful math communication? What if our learners had a learning progression that offered the opportunity to level up in math communication?

What if it looked like this?

Level 4
I can show more than one way to find a solution to the problem. I can choose appropriately from writing a complete sentence, drawing a picture, writing a number algorithm, or another creative way.

Level 3
I can find a solution to the problem and describe or illustrate how I arrived at the solution in a way that the reader does not have to talk with me in person to understand my path to the solution.

Level 2
I can find a correct solution to the problem.

Level 1
I can ask questions to help me work toward a solution to the problem.

What if this became a norm? What if we used this or something similar to help our learners self-assess their mathematical written communication? If we emphasize math communication at this early age, will we ultimately have more confident and communicative math students in middle school and high school?

What if we lead learners to level up in communication of understanding? What if we take up the challenge to make thinking visible? … to show what we know more than one way? … to communicate where the reader doesn’t have to ask questions?

A doodler is connecting neurological pathways with previously disconnected pathways. A doodler is concentrating intently, sifting though information, conscious, and otherwise, and – much more often than we realize – generating massive insights. (Brown, 11 pag.)

How might we practice, experience, and engage in a different way of connecting with information? What if we exercise our own creativity to create visuals of what we are learning?

Rather than diverting our attention away from a topic (what our culture believe is happening when people doodle), doodling can serve as an anchoring task – a task that can occur simultaneously with another task – and act as a preemptive measure to keep us from losing focus on [a] topic. (Brown, 18 pag.)

It seems counterintuitive, but I can attest to my own improvement in focus, attention, and engagement.

People using even rudimentary visual langue to understand or express something are stirring the neurological pathways of the mind to see a topic in a new light. (Brown, 71 pag.)

How might we foster a community of learners where everyone bravely and fiercely seeks feedback?

I was at EduCon in Philadelphia when this tweet arrived last week.

Am I showing enough work? How do I know? What if we partner, students and teachers, to seek feedback, clarity, and guidance?

Success inspires success.

Yesterday, I dropped by Kato‘s classroom to work on the next math assessment and found our learners working together to apply math and to improve communication.

Now, I was just sneaking in to drop off and pick up papers. But, how could I turn down requests for feedback?

Here’s the #showyourwork #LL2LU progression in the classroom:

Grade 4

Level 4
I can show more than one way to find a solution to the problem.

Level 3I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2
I can find a correct solution to the problem.

Level 1
I can ask questions to help me work toward a solution to the problem.

And here’s one child and her work. “Ms. Gough, will you look at my work? Can you understand it without asking me questions? Is is clear to you?”

I see connected words, pictures, and numbers. I like the color coding for the different size bags. I appreciate reading the sentences that explain the numbers and her thinking. I also witnessed this young learner improve her work and her thinking while watching me read her work. She knew what she wanted to add, because she wished I knew why she made the final choice. I’d call this Level 4 work.

What if we foster a community of learners who bravely and fiercely seek feedback?